If you have Adobe Reader installed, opened PDFs ignore the Windows settings that MattB mentions. This is because Adobe install an ActiveX control into Internet Explorer that changes the open behaviour.

You need to check the preferences of Adobe Reader to see if the setting can be disabled.

IIRC, you may need to modify the Security Settings in IE for the specific zone that the site is in where you're downloading the PDF (I'm assuming Local Intranet Zone for Sharepoint). Tools->Internet Options->Security tab->Select zone->Custom level, and then scroll down to "ActiveX controls and plug-ins," and you will have to Enable or Disable some of the prompting/downloading items.

+1; always check the zone first and foremost. Check what version of IE your clients are using too, as the security model has changed a bit recently.
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Darth SatanAug 26 '09 at 18:03

It's amazing that no-one got deep enough to detail that there's a security setting in the default "Medium" and lower trust levels that sets "Launching applications and unsafe files" to "Prompt", while "Local intranet" by default uses the default "Medium-low" trust level, where this setting is set to "Enable". Simply listing your domain in the Local intranet should solve the issue.
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Alexey KrasheninnikovJul 9 '12 at 19:55

In some companies the Trusted sites are considered less secure than Local intranet and use the default Medium security level which forces the browser to continue prompting despite the site being trusted. See my answer.
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Alexey KrasheninnikovJul 9 '12 at 20:04

This issue does not depend on any version of SharePoint whatsoever and is not primarily related to SharePoint either. It is a browser security zone setting and dialog box enhancement of Internet Explorer since IE6 SP1 described here: KB 821968.

Yes, you can disable the browser add-on breaking the create and edit new Office documents functionality of SharePoint aka Client Integration.

Yes, you can fiddle with File Types settings, but in the end all of these solutions help breach the security of your PC instead of accurately solving the issue where it can be solved more legitimately.

It's amazing that no-one got deep enough to detail that there's a security setting in the default "Medium" and lower security levels that sets "Launching applications and unsafe files" to "Prompt", while "Local intranet" by default uses the default "Medium-low" security level, where this setting is set to "Enable".

Simply listing your domain in the Local intranet instead of Trusted Sites should solve the issue described here as well as some of the SSO issues out there automatically.